The present invention relates to a process for the noncoherent demodulation of a linearly modulated signal where each symbol .alpha..sub.k has the same energy and a demodulator for performing this process. It is more particularly used in satellite links and in vehicle radio communications.
A digital signal x(t), in which t is a time variable, is generally a narrow band signal centered about a frequency f.sub.0 called the carrier frequency and of band width B. It is therefore a signal, whose spectral density is zero outside the frequency spacing, [f.sub.0 -B/2, f.sub.0 +B/2].
In signal theory, it is standard practice to represent this digital signal by its complex envelope .alpha.(t), the relation between x(t) and .alpha.(t) being given by the equation EQU x(t)=Re(.alpha.(t).e.sup.jw.sbsp.0.sup.t)
in which Re signifies "real part of", e signifies exponential and j is a complex number, such as j.sup.2 =-1 and w.sub.0 is the ripple corresponding to the frequency f.sub.0. As the complex representation permits a clearer definition, it will be used throughout the remainder of the text.
Consideration will be given to a sequence of N symbols a.sub.0, . . . , a.sub.k, . . . , a.sub.N-1 where k is an integer and where each a.sub.k represents an information to be transmitted, where N equals approximately 64. These symbols are elements of an e.g. binary alphabet A. For minimizing the error rate in transmitting the sequence of symbols a.sub.0 . . . a.sub.N-1, it is standard practice to encode this sequence into another sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1 in which each symbol of said other sequence belongs to another alphabet .alpha., which is e.g. of the M-type, in which M is an integer.
This sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1 will effectively be transmitted by a modulated signal, the modulation being realized by the said signals. In the case of a linear modulation, the complex envelope .alpha.(t) of the modulated signal x(t) containing the information to be transmitted is then represented by the expression ##EQU1## in which T is the time interval between the transmission of two successive symbols and g(t) is a function, with real or complex values, describing the pulse response of all the emission filters so that .vertline..alpha..sub.k .vertline..sup.2 =1 and ##EQU2## dt, equal to E.sub.b is the energy per bit.
FIG. 1 is a diagrammatic representation illustrating the known chain of modulation, transmission and demodulation of symbols a.sub.0 . . . a.sub.N-1, which are sequentially received in a modulator 2. They follow one another spaced by a time interval T. The modulator 2 comprises a coding means 4 supplying at the output the sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1, which are spaced from one another by a time interval T. It also comprises a modulation means 6, which supplies the complex envelope .alpha.(t) in the form of its real part, Re(.alpha.(t)) and its imaginary part Im(.alpha.(t)). These two signals are frequency inverted by respectively modulating a signal cos (w.sub.0 t) supplied by an oscillator 8 and a signal -sin (w.sub.0 t) supplied by a phase shifter 10, which is connected by the input to oscillator 8. The two resulting modulated signals are summated and their sum constitutes the emitted signal x(t).
This emitted signal x(t) during transmission, is subject to disturbances, represented by the addition of a Gaussian white noise b(t) of bilateral spectral density N.sub.0 /2 in watt/hertz. Thus, demodulator 12 receives a signal y(t) equal to x(t)+b(t), which is frequency reinverted by modulating a first signal 2. cos (w.sub.0.t+.theta.(t)) from an oscillator 14 and a second signal -2. sin (w.sub.0.t+.theta.(t)) supplied by a phase shifter 16, which is connected to the same oscillator 14. The phase .theta.(t) of the signals is now known in the case of a non-coherent demodulation, but its variation is slow compared with the binary flow rate of transmission. The modulated signals are respectively designated Re(r(t)) and Im(r(t)). These are the real and imaginary components of the complex envelope r(t) of the signal y(t). These signals are filtered by a matched filtering means 18 of pulse response g(t.sub.0 -t), in which t.sub.0 is a quantity characterizing the transmission time of the signal all along the chain. A means 20 then samples these signals and supplies at the output the real part Re(r.sub.k ) and imaginary part Im(r.sub.k) of the observation r.sub.k in which 0.ltoreq.k.ltoreq.N-1. This means 20 performs a sampling at dates separated by a time interval T.
The observations r.sub.0 . . . r.sub.N-1 are sequentially applied to the input of a calculating or computing means 22, which supplies at the output a sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1 which represent the most probable estimated values of the symbols .alpha..sub.0 . . . .alpha..sub.N-1 emitted, bearing in mind the observations r.sub.0 . . . r.sub.N-1. If the transmission of the symbols .alpha..sub.0 . . . .alpha..sub.N-1 is perfect, the symbols .alpha..sub.0 . . . .alpha..sub.N-1 are respectively identical to said symbols .alpha..sub.0 . . . .alpha..sub.N-1. The sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1 obtained is then decoded by a decoding means 24, which supplies a sequence of symbols a.sub.0 . . . a.sub.N-1. The latter are respectively identical to the symbols a.sub.0 . . . a.sub.N-1, if the transmission is perfect.
The demodulation performed is said to be noncoherent if the phase .theta.(t) of the signal emitted by oscillator 14 is not known. This is particularly the case if this oscillator is free, i.e. if it is not dependent on the signal received y(t).
In general terms, the demodulation, i.e. the determination of the most probable sequence of symbols .alpha..sub.0 . . . .alpha..sub.N-1 minimizes the error rate on reception, if the symbols emitted are equiprobable. It is known from the article "Optical Reception of Digital Data over the Gaussian Channel with Unknown Delay and Phase Jitter" by David D. Falconer, which appeared in IEEE Transactions on Information Theory, January 1977, pp. 117-126, that the probability function in coherent reception maximizes: ##EQU3## as a function of the symbols .alpha..sub.0 . . . .alpha..sub.N-1, in which r*(t) is the conjugate complex of r(t) and in which .theta.(t) characterized the phase introduced by the transmission channel.
This maximization cannot be performed in the case of a coherent demodulation in which .theta.(t) is a known function of the receiver, which can thus be assumed as zero. In the case of a noncoherent demodulation, it can be maximized if it assumed, and this is a reasonable hypothesis, that .theta.(t) is constant over the time interval [0,NT] and is considered by the receiver as a random variable equally distributed on [0,2.pi.] said constant being unknown to the receiver. It is known that the quantity maximized is then ##EQU4##
By replacing .alpha.(t, .alpha..sub.0, . . . , .alpha..sub.N-1) by ##EQU5## and on noting that the square of the absolute value of a complex number is equal to the product of this complex number by its conjugate, it is possible to replace the quantity to be maximized by the equivalent quantity ##EQU6##
The receiver or demodulator determining the sequence of symbols .alpha..sub.0 . . . , .alpha..sub.N-1 maximizing this quantity is said to be optimum in the sense of the probability maximum. In practice, the maximization is of a rising complexity with N (the calculation number rises as 2.sup.N) and for N higher than about 10, it is not known how this expression can be solved in a simple manner.
Several methods are known which make it possible to obtain a suboptimum receiver. It is possible to choose an observation window formed by a single symbol (N=1). This leads to the conventional noncoherent receiver, more particularly used in low speed modems, which observe the signal symbol by symbol.
A noncoherent receiver using a two symbol observation window is also known. in the case of phase shift keying (PSK), this receiver is the conventional differential receiver.
A noncoherent receiver using an observation window of two 2P+1 symbols, in which P is an integer, is known for determining the central symbol. This method described in the article "Coherent and noncoherent detection of CPDSK" by W. P. Osborne and M. B. Luntz, which appeared in IEEE Transactions on Communications, vol. COM-22, no. 8, August 1974, pp. 1023-1036 leads to a receiver which cannot be produced, because the probability function is not expressed in a simpler manner as a function of the observation.